There's a question that asks for the 2nd derivative of y−2x−3xy=2

Abdiel Mays 2022-11-20 Answered
Please excuse if the formatting of this post is wrong.

There's a question that asks for the 2nd derivative of y 2 x 3 x y = 2

From what I know, I have to use implicit differentiation, using which I get:
12 + 18 y ( 1 3 x ) 2
But can you solve for y in the initial equation and differentiate two times (aka explicit differentiation)? By doing that I got:
48 ( 1 3 x ) 3
I'm not sure if this is a correct answer as I am new to differentiation. I guess that this question is also tied with another question; can you substitute y in implicit differentiation by solving for it in the initial equation?
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Answers (2)

mainzollbtt
Answered 2022-11-21 Author has 13 answers
Sure you can in this case
y = 2 ( x + 1 ) 1 3 x
and then
y = 8 ( 1 3 x ) 2
and finally
y = 48 ( 1 3 x ) 3
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Elliana Molina
Answered 2022-11-22 Author has 3 answers
We differentiate once,
y 2 3 y 3 x y = 0
and twice,
y 3 y 3 y 3 x y = 0.
Now we can eliminate y , using
( 1 3 x ) y = 3 y + 2
and
( 1 3 x ) y = 6 y = 6 3 y + 2 1 3 x .
You can also eliminate y using the initial equation.
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